Form of tension minima and resulting accuracy in

As discussed in Section 5.3.1 (also see Appendix J),
the values of these minima at each is determined
by the quality of the Helmholtz
basis set.
Around each minimum the tension is quadratic,

(5.23)

where the curvatures are very similar for all states
(see Fig. 5.2).
This happens because the lowest tension state of constant norm (that is, the
state returned by (5.14)) is simply the
scaling eigenfunction with lowest tension
(see Section 6.1.1).
The tension expansion of this function about is quadratic with
exactly, independent of , for a modified weight function
(see Appendix I).
Therefore with our choice
, a curvature estimate is
where the averages are
taken over
, and
.
If either there is little scarring or the billiard is close to spherical,
then we expect this estimate to be good.

Eq.(5.23) defines a natural `error scale' for the found:
a change that is much less than
from a
tension minimum
causes only a small fractional change in tension.
Therefore I will call
the `tension rounding error' of the
state , because it defines an uncertainty in due to the
basis-dependent finite tension minima.
In practice, the errors in found appear to be an order of magnitude better
than
, as shwon by the lines in Fig. 6.6.